* May advised by some to call an election (Updates after motions are selected, adds sterling)

By Elizabeth Piper, Kylie MacLellan and William James

LONDON, April 1 (Reuters) - Lawmakers sought to break a stalemate over Britain's departure from the European Union on Monday by trying to force Prime Minister Theresa May to pursue much closer economic ties than the deal she negotiated envisages.

After a tumultuous few weeks in which May's divorce strategy was rejected by lawmakers for a third time, despite her offer to quit if it passed, the future direction of Brexit remains mired in confusion and acrimony.

Three days after the date on which Britain was originally due to leave the EU, it is still uncertain how, when or even whether the United Kingdom will say goodbye to the bloc it joined 46 years ago.

In a bid to break the impasse, lawmakers took control of parliamentary proceedings and were set at vote at 8 p.m. (1900 GMT) on four options ranging from staying in the EU's single market to revoking the formal divorce papers to avoid a no-deal exit.

The third defeat of May's withdrawal agreement on Friday left one of the weakest British leaders in a generation facing a spiralling crisis over Brexit, the United Kingdom's most far-reaching policy change since World War Two.

May's government and her Conservative party, which has been trying to contain a schism over Europe for 30 years, are in open conflict between those pushing for a customs union with the EU and eurosceptics demanding a cleaner break with the bloc.

May's chief whip, responsible for party discipline, told the BBC in a documentary being aired on Monday that the government should have known that May's loss of her parliamentary majority in a snap election in 2017 would "inevitably" lead to a softer Brexit.

The issue has pushed aside almost all others - a point made by a dozen semi-naked climate change protesters who stood against a glass panel in a public gallery overlooking the chamber as lawmakers debated.

Beyond Westminster, the UK head of German industrial giant Siemens, Juergen Maier, said Britain was wrecking its reputation, and urged lawmakers to back a customs union with the EU.

"We are all running out of patience," Maier said. "Where the UK used to be beacon for stability, we are now becoming a laughing stock," he said in an open letter to lawmakers published by the website Politico.

ASK THE SPHINX

From EU officials watching from Brussels, there was one plea - make up your minds.

"A sphinx is an open book compared to the UK," said Jean-Claude Juncker, European Commission president. "Nobody knows where it is heading. Would like to make the sphinx talk and tell us in which direction they would like to go."

In the 2016 Brexit referendum, 17.4 million voters, or 51.9 percent, backed leaving the EU while 16.1 million, or 48.1 percent, backed staying.

But ever since, opponents have sought to soften, or even stop, the divorce, while proponents have failed to unite around the deal May negotiated with the EU.

Parliament was due to vote on four alternative Brexit options selected by the speaker, John Bercow.

These include remaining in a customs union with the EU; membership of the single market with a customs arrangement; putting any agreed deal to a second referendum; and halting Brexit altogether if necessary to avoid leaving the bloc without a deal.

May is considering a final throw of the dice by bringing her deal back to a vote in parliament as soon as Wednesday, when it could be put to lawmakers in a "run-off" against whichever alternative gains the most support.

Sterling rose on investor hopes that May would be forced to go for a softer Brexit than planned.

However, several lawmakers said they saw a national election as the only way to find a solution.

Eurosceptic lawmaker Steve Baker, a member of May's Conservative Party, suggested that any backing for a customs union could push him to support a move to topple the government.

Britain had been due to leave the EU on March 29 but the political deadlock in London forced May to ask the bloc for a delay. As things stand, Britain will now depart at 2200 GMT on April 12 - unless May comes up with another option.